Indices |
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An index (or exponent) tells how many times a number is multiplied by itself. For example, 34 = 3 × 3 × 3 × 3 = 81.
A root is the inverse of an index. For example, the square root of 81 is √81 = 9.
In this topic you will explore powers, roots, and the rules that connect them — known as the Laws of Indices.
Rule: a0 = 1 (for any a ≠ 0).
A non-zero base raised to the power of zero is always 1.
Example: 50 = 1.
Rule: a1 = a.
A base raised to the power of one is itself.
Example: 71 = 7.
Rule: am × an = am + n.
When multiplying exponential terms with the same base, add the exponents.
Example: x2 × x3 = x2+3 = x5.
Rule: (am) ÷ (an) = am − n (for a ≠ 0).
When dividing exponential terms with the same base, subtract the exponents.
Example: (y5) ÷ (y2) = y5−2 = y3.
Rule: (am)n = amn.
When raising a power to another power, multiply the exponents.
Example: (x2)3 = x2 × 3 = x6.
Rule: a−n = 1/(an) (for a ≠ 0).
A negative exponent indicates the reciprocal of the base raised to the corresponding positive power.
Example: 2−3 = 1/(23) = 1/8.
Scientific notation is a way of writing very large or very small numbers using powers of 10. It helps us write numbers clearly and compare their size more easily.
In scientific notation a number is written in the form a × 10n, where 1 ≤ a < 10 and n is an integer.
For example:
